Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.9%

Time: 10.8s

Alternatives: 9

Speedup: 2.6×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (/
    (+
     (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0)))
     (fma 0.6666666666666666 (* x x) 2.0))
    (sqrt PI)))))

C

double code(double x) {
	return fabs(x) * fabs(((fma(0.2, pow(x, 4.0), (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(fma(0.2, (x ^ 4.0), Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))))
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 34.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fabs
   (*
    (pow PI -0.5)
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0)))))))))

C

double code(double x) {
	return x * fabs((pow(((double) M_PI), -0.5) * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0)))))));
}

Java

public static double code(double x) {
	return x * Math.abs((Math.pow(Math.PI, -0.5) * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))))));
}

Python

def code(x):
	return x * math.fabs((math.pow(math.pi, -0.5) * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * math.pow(x, 2.0)))))))

Julia

function code(x)
	return Float64(x * abs(Float64((pi ^ -0.5) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * abs(((pi ^ -0.5) * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0)))))));
end

Wolfram

code[x_] := N[(x * N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 28.7%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   2. fabs-sqr 28.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   3. add-sqr-sqrt 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   4. *-un-lft-identity 30.3%

      \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

7. Applied egg-rr 30.3%

   \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
8. Step-by-step derivation

   1. *-lft-identity 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

9. Simplified 30.3%

   \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
10. Step-by-step derivation

    1. *-un-lft-identity 30.3%

       \[\leadsto x \cdot \left|\color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    2. pow1/2 30.3%

       \[\leadsto x \cdot \left|\left(1 \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    3. inv-pow 30.3%

       \[\leadsto x \cdot \left|\left(1 \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    4. pow-pow 30.3%

       \[\leadsto x \cdot \left|\left(1 \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    5. metadata-eval 30.3%

       \[\leadsto x \cdot \left|\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

11. Applied egg-rr 30.3%

    \[\leadsto x \cdot \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. *-lft-identity 30.3%

       \[\leadsto x \cdot \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

13. Simplified 30.3%

    \[\leadsto x \cdot \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
14. Add Preprocessing

Alternative 3: 34.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fabs
   (/
    (+
     (* 0.047619047619047616 (pow x 6.0))
     (fma 0.6666666666666666 (* x x) 2.0))
    (sqrt PI)))))

C

double code(double x) {
	return x * fabs((((0.047619047619047616 * pow(x, 6.0)) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(x * abs(Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))))
end

Wolfram

code[x_] := N[(x * N[Abs[N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.2%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. add-sqr-sqrt 28.7%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   2. fabs-sqr 28.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   3. add-sqr-sqrt 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   4. *-un-lft-identity 30.3%

      \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

7. Applied egg-rr 30.3%

   \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
8. Step-by-step derivation

   1. *-lft-identity 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

9. Simplified 30.3%

   \[\leadsto \color{blue}{x} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
10. Add Preprocessing

Alternative 4: 34.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fabs (/ (+ (* 0.047619047619047616 (pow x 6.0)) 2.0) (sqrt PI)))))

C

double code(double x) {
	return x * fabs((((0.047619047619047616 * pow(x, 6.0)) + 2.0) / sqrt(((double) M_PI))));
}

Java

public static double code(double x) {
	return x * Math.abs((((0.047619047619047616 * Math.pow(x, 6.0)) + 2.0) / Math.sqrt(Math.PI)));
}

Python

def code(x):
	return x * math.fabs((((0.047619047619047616 * math.pow(x, 6.0)) + 2.0) / math.sqrt(math.pi)))

Julia

function code(x)
	return Float64(x * abs(Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + 2.0) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = x * abs((((0.047619047619047616 * (x ^ 6.0)) + 2.0) / sqrt(pi)));
end

Wolfram

code[x_] := N[(x * N[Abs[N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.2%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. add-sqr-sqrt 28.7%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   2. fabs-sqr 28.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   3. add-sqr-sqrt 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   4. *-un-lft-identity 30.3%

      \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

7. Applied egg-rr 30.3%

   \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
8. Step-by-step derivation

   1. *-lft-identity 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

9. Simplified 30.3%

   \[\leadsto \color{blue}{x} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

10. Taylor expanded in x around 0 30.0%

    \[\leadsto x \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Alternative 5: 98.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left|{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (pow PI -0.5)
   (+
    (* x 2.0)
    (* 0.047619047619047616 (* (* x x) (* (* x x) (* x (* x x)))))))))

C

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Java

public static double code(double x) {
	return Math.abs((Math.pow(Math.PI, -0.5) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Python

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))))

Julia

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(Float64(x * 2.0) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((pi ^ -0.5) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. rem-square-sqrt 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. rem-square-sqrt 98.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Simplified 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
8. Step-by-step derivation

   1. add-sqr-sqrt 28.7%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   2. fabs-sqr 28.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   3. add-sqr-sqrt 30.3%

      \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

   4. *-un-lft-identity 30.3%

      \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

9. Applied egg-rr 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
10. Step-by-step derivation

    1. *-lft-identity 30.3%

       \[\leadsto \color{blue}{x} \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

11. Simplified 98.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. *-un-lft-identity 98.8%

       \[\leadsto \left|\color{blue}{\left(1 \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    2. inv-pow 98.8%

       \[\leadsto \left|\left(1 \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{-1}}\right) \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    3. sqrt-pow2 98.8%

       \[\leadsto \left|\left(1 \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right) \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    4. metadata-eval 98.8%

       \[\leadsto \left|\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

13. Applied egg-rr 98.8%

    \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
14. Step-by-step derivation

    1. *-lft-identity 98.8%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

15. Simplified 98.8%

    \[\leadsto \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

16. Final simplification 98.8%

    \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
17. Add Preprocessing

Alternative 6: 34.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (pow PI -0.5) (* x 2.0))
   (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = (0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = (0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = (0.047619047619047616 * (x ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
   6. Step-by-step derivation

      1. rem-square-sqrt 28.4%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

      2. fabs-sqr 28.4%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

      3. rem-square-sqrt 98.8%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. Simplified 98.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   8. Taylor expanded in x around 0 67.5%

      \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   9. Step-by-step derivation

      1. *-commutative 67.5%

         \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

      2. associate-*r* 67.5%

         \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

      3. unpow-1 67.5%

         \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right)\right| \]

      4. metadata-eval 67.5%

         \[\leadsto \left|x \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot 2\right)\right| \]

      5. pow-sqr 67.5%

         \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot 2\right)\right| \]

      6. rem-sqrt-square 67.5%

         \[\leadsto \left|x \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot 2\right)\right| \]

      7. rem-square-sqrt 67.5%

         \[\leadsto \left|x \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot 2\right)\right| \]

      8. fabs-sqr 67.5%

         \[\leadsto \left|x \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot 2\right)\right| \]

      9. rem-square-sqrt 67.5%

         \[\leadsto \left|x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right)\right| \]

   10. Simplified 67.5%

       \[\leadsto \left|\color{blue}{x \cdot \left({\pi}^{-0.5} \cdot 2\right)}\right| \]

   11. Taylor expanded in x around 0 67.5%

       \[\leadsto \color{blue}{\left|2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
   12. Step-by-step derivation

       1. associate-*r* 67.5%

          \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

       2. fabs-mul 67.5%

          \[\leadsto \color{blue}{\left|2 \cdot x\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right|} \]

       3. *-commutative 67.5%

          \[\leadsto \left|\color{blue}{x \cdot 2}\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       4. rem-exp-log 67.5%

          \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

       5. exp-neg 67.5%

          \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\color{blue}{e^{-\log \pi}}}\right| \]

       6. unpow1/2 67.5%

          \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}}\right| \]

       7. exp-prod 67.5%

          \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}}\right| \]

       8. distribute-lft-neg-out 67.5%

          \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{-\log \pi \cdot 0.5}}\right| \]

       9. distribute-rgt-neg-in 67.5%

          \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}}\right| \]

       10. metadata-eval 67.5%

           \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\log \pi \cdot \color{blue}{-0.5}}\right| \]

       11. exp-to-pow 67.5%

           \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\pi}^{-0.5}}\right| \]

       12. fabs-mul 67.5%

           \[\leadsto \color{blue}{\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|} \]

       13. associate-*r* 67.5%

           \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}\right| \]

       14. rem-square-sqrt 28.4%

           \[\leadsto \left|\color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}}\right| \]

       15. fabs-sqr 28.4%

           \[\leadsto \color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

       16. rem-square-sqrt 30.1%

           \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

       17. associate-*r* 30.1%

           \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot {\pi}^{-0.5}} \]

   13. Simplified 29.9%

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}} \]
   14. Step-by-step derivation

       1. div-inv 30.1%

          \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{\sqrt{\pi}}} \]

       2. inv-pow 30.1%

          \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{-1}} \]

       3. sqrt-pow2 30.1%

          \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \]

       4. metadata-eval 30.1%

          \[\leadsto \left(2 \cdot x\right) \cdot {\pi}^{\color{blue}{-0.5}} \]

       5. *-commutative 30.1%

          \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

       6. associate-*r* 30.1%

          \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

   15. Applied egg-rr 30.1%

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
   16. Step-by-step derivation

       1. *-commutative 30.1%

          \[\leadsto \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right) \cdot x} \]

       2. *-commutative 30.1%

          \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot 2\right)} \cdot x \]

       3. associate-*r* 30.1%

          \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(2 \cdot x\right)} \]

   17. Simplified 30.1%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(2 \cdot x\right)} \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. *-commutative 98.9%

         \[\leadsto \left|\color{blue}{\left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \left|x\right|\right)} \cdot \sqrt{\frac{1}{\pi}}\right| \]

      3. rem-square-sqrt 28.4%

         \[\leadsto \left|\left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      4. fabs-sqr 28.4%

         \[\leadsto \left|\left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      5. rem-square-sqrt 98.9%

         \[\leadsto \left|\left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \color{blue}{x}\right) \cdot \sqrt{\frac{1}{\pi}}\right| \]

      6. associate-*l* 98.9%

         \[\leadsto \left|\color{blue}{\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   7. Simplified 98.9%

      \[\leadsto \left|\color{blue}{\left(2 + {x}^{4} \cdot \mathsf{fma}\left({x}^{2}, 0.047619047619047616, 0.2\right)\right) \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   8. Taylor expanded in x around inf 37.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   9. Step-by-step derivation

      1. *-commutative 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)}\right| \]

      2. unpow-1 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot {x}^{7}\right)\right| \]

      3. metadata-eval 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot {x}^{7}\right)\right| \]

      4. pow-sqr 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot {x}^{7}\right)\right| \]

      5. rem-sqrt-square 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot {x}^{7}\right)\right| \]

      6. rem-square-sqrt 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot {x}^{7}\right)\right| \]

      7. fabs-sqr 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot {x}^{7}\right)\right| \]

      8. rem-square-sqrt 37.0%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot {x}^{7}\right)\right| \]

   10. Simplified 37.0%

       \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)}\right| \]

   11. Taylor expanded in x around 0 37.0%

       \[\leadsto \color{blue}{\left|0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
   12. Step-by-step derivation

       1. associate-*r* 36.9%

          \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

       2. fabs-mul 36.9%

          \[\leadsto \color{blue}{\left|0.047619047619047616 \cdot {x}^{7}\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right|} \]

       3. fabs-mul 36.9%

          \[\leadsto \color{blue}{\left(\left|0.047619047619047616\right| \cdot \left|{x}^{7}\right|\right)} \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       4. metadata-eval 36.9%

          \[\leadsto \left(\color{blue}{0.047619047619047616} \cdot \left|{x}^{7}\right|\right) \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       5. metadata-eval 36.9%

          \[\leadsto \left(0.047619047619047616 \cdot \left|{x}^{\color{blue}{\left(2 \cdot 3.5\right)}}\right|\right) \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       6. pow-sqr 1.6%

          \[\leadsto \left(0.047619047619047616 \cdot \left|\color{blue}{{x}^{3.5} \cdot {x}^{3.5}}\right|\right) \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       7. fabs-sqr 1.6%

          \[\leadsto \left(0.047619047619047616 \cdot \color{blue}{\left({x}^{3.5} \cdot {x}^{3.5}\right)}\right) \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       8. pow-sqr 3.6%

          \[\leadsto \left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(2 \cdot 3.5\right)}}\right) \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       9. metadata-eval 3.6%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right) \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

       10. rem-exp-log 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|\sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

       11. exp-neg 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|\sqrt{\color{blue}{e^{-\log \pi}}}\right| \]

       12. unpow1/2 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}}\right| \]

       13. exp-prod 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}}\right| \]

       14. distribute-lft-neg-out 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|e^{\color{blue}{-\log \pi \cdot 0.5}}\right| \]

       15. distribute-rgt-neg-in 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}}\right| \]

       16. metadata-eval 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|e^{\log \pi \cdot \color{blue}{-0.5}}\right| \]

       17. exp-to-pow 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|\color{blue}{{\pi}^{-0.5}}\right| \]

       18. metadata-eval 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

       19. pow-sqr 3.6%

           \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \]

   13. Simplified 3.6%

       \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.7% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow PI -0.5) (* x 2.0)))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * (x * 2.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * (x * 2.0);
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * (x * 2.0)

Julia

function code(x)
	return Float64((pi ^ -0.5) * Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * (x * 2.0);
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. rem-square-sqrt 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. rem-square-sqrt 98.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Simplified 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Taylor expanded in x around 0 67.5%

   \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
9. Step-by-step derivation

   1. *-commutative 67.5%

      \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

   2. associate-*r* 67.5%

      \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

   3. unpow-1 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right)\right| \]

   4. metadata-eval 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot 2\right)\right| \]

   5. pow-sqr 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot 2\right)\right| \]

   6. rem-sqrt-square 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot 2\right)\right| \]

   7. rem-square-sqrt 67.5%

      \[\leadsto \left|x \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot 2\right)\right| \]

   8. fabs-sqr 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot 2\right)\right| \]

   9. rem-square-sqrt 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right)\right| \]

10. Simplified 67.5%

    \[\leadsto \left|\color{blue}{x \cdot \left({\pi}^{-0.5} \cdot 2\right)}\right| \]

11. Taylor expanded in x around 0 67.5%

    \[\leadsto \color{blue}{\left|2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
12. Step-by-step derivation

    1. associate-*r* 67.5%

       \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

    2. fabs-mul 67.5%

       \[\leadsto \color{blue}{\left|2 \cdot x\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right|} \]

    3. *-commutative 67.5%

       \[\leadsto \left|\color{blue}{x \cdot 2}\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

    4. rem-exp-log 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

    5. exp-neg 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\color{blue}{e^{-\log \pi}}}\right| \]

    6. unpow1/2 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}}\right| \]

    7. exp-prod 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}}\right| \]

    8. distribute-lft-neg-out 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{-\log \pi \cdot 0.5}}\right| \]

    9. distribute-rgt-neg-in 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}}\right| \]

    10. metadata-eval 67.5%

        \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\log \pi \cdot \color{blue}{-0.5}}\right| \]

    11. exp-to-pow 67.5%

        \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\pi}^{-0.5}}\right| \]

    12. fabs-mul 67.5%

        \[\leadsto \color{blue}{\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|} \]

    13. associate-*r* 67.5%

        \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}\right| \]

    14. rem-square-sqrt 28.4%

        \[\leadsto \left|\color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}}\right| \]

    15. fabs-sqr 28.4%

        \[\leadsto \color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

    16. rem-square-sqrt 30.1%

        \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

    17. associate-*r* 30.1%

        \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot {\pi}^{-0.5}} \]

13. Simplified 29.9%

    \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}} \]
14. Step-by-step derivation

    1. div-inv 30.1%

       \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{\sqrt{\pi}}} \]

    2. inv-pow 30.1%

       \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{{\left(\sqrt{\pi}\right)}^{-1}} \]

    3. sqrt-pow2 30.1%

       \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \]

    4. metadata-eval 30.1%

       \[\leadsto \left(2 \cdot x\right) \cdot {\pi}^{\color{blue}{-0.5}} \]

    5. *-commutative 30.1%

       \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

    6. associate-*r* 30.1%

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

15. Applied egg-rr 30.1%

    \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
16. Step-by-step derivation

    1. *-commutative 30.1%

       \[\leadsto \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right) \cdot x} \]

    2. *-commutative 30.1%

       \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot 2\right)} \cdot x \]

    3. associate-*r* 30.1%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(2 \cdot x\right)} \]

17. Simplified 30.1%

    \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(2 \cdot x\right)} \]

18. Final simplification 30.1%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot 2\right) \]
19. Add Preprocessing

Alternative 8: 34.7% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(x \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (* x (pow PI -0.5))))

C

double code(double x) {
	return 2.0 * (x * pow(((double) M_PI), -0.5));
}

Java

public static double code(double x) {
	return 2.0 * (x * Math.pow(Math.PI, -0.5));
}

Python

def code(x):
	return 2.0 * (x * math.pow(math.pi, -0.5))

Julia

function code(x)
	return Float64(2.0 * Float64(x * (pi ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x * (pi ^ -0.5));
end

Wolfram

code[x_] := N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. rem-square-sqrt 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. rem-square-sqrt 98.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Simplified 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Taylor expanded in x around 0 67.5%

   \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
9. Step-by-step derivation

   1. *-commutative 67.5%

      \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

   2. associate-*r* 67.5%

      \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

   3. unpow-1 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right)\right| \]

   4. metadata-eval 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot 2\right)\right| \]

   5. pow-sqr 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot 2\right)\right| \]

   6. rem-sqrt-square 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot 2\right)\right| \]

   7. rem-square-sqrt 67.5%

      \[\leadsto \left|x \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot 2\right)\right| \]

   8. fabs-sqr 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot 2\right)\right| \]

   9. rem-square-sqrt 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right)\right| \]

10. Simplified 67.5%

    \[\leadsto \left|\color{blue}{x \cdot \left({\pi}^{-0.5} \cdot 2\right)}\right| \]

11. Taylor expanded in x around 0 67.5%

    \[\leadsto \color{blue}{\left|2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
12. Step-by-step derivation

    1. associate-*r* 67.5%

       \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

    2. fabs-mul 67.5%

       \[\leadsto \color{blue}{\left|2 \cdot x\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right|} \]

    3. *-commutative 67.5%

       \[\leadsto \left|\color{blue}{x \cdot 2}\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

    4. rem-exp-log 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

    5. exp-neg 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\color{blue}{e^{-\log \pi}}}\right| \]

    6. unpow1/2 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}}\right| \]

    7. exp-prod 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}}\right| \]

    8. distribute-lft-neg-out 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{-\log \pi \cdot 0.5}}\right| \]

    9. distribute-rgt-neg-in 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}}\right| \]

    10. metadata-eval 67.5%

        \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\log \pi \cdot \color{blue}{-0.5}}\right| \]

    11. exp-to-pow 67.5%

        \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\pi}^{-0.5}}\right| \]

    12. fabs-mul 67.5%

        \[\leadsto \color{blue}{\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|} \]

    13. associate-*r* 67.5%

        \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}\right| \]

    14. rem-square-sqrt 28.4%

        \[\leadsto \left|\color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}}\right| \]

    15. fabs-sqr 28.4%

        \[\leadsto \color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

    16. rem-square-sqrt 30.1%

        \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

13. Simplified 30.1%

    \[\leadsto \color{blue}{2 \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]
14. Add Preprocessing

Alternative 9: 34.5% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 2.0) (sqrt PI)))

C

double code(double x) {
	return (x * 2.0) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (x * 2.0) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (x * 2.0) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(x * 2.0) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. rem-square-sqrt 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 28.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. rem-square-sqrt 98.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot \color{blue}{x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Simplified 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot x} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Taylor expanded in x around 0 67.5%

   \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
9. Step-by-step derivation

   1. *-commutative 67.5%

      \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

   2. associate-*r* 67.5%

      \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

   3. unpow-1 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 2\right)\right| \]

   4. metadata-eval 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot 2\right)\right| \]

   5. pow-sqr 67.5%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot 2\right)\right| \]

   6. rem-sqrt-square 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot 2\right)\right| \]

   7. rem-square-sqrt 67.5%

      \[\leadsto \left|x \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot 2\right)\right| \]

   8. fabs-sqr 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot 2\right)\right| \]

   9. rem-square-sqrt 67.5%

      \[\leadsto \left|x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right)\right| \]

10. Simplified 67.5%

    \[\leadsto \left|\color{blue}{x \cdot \left({\pi}^{-0.5} \cdot 2\right)}\right| \]

11. Taylor expanded in x around 0 67.5%

    \[\leadsto \color{blue}{\left|2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right|} \]
12. Step-by-step derivation

    1. associate-*r* 67.5%

       \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

    2. fabs-mul 67.5%

       \[\leadsto \color{blue}{\left|2 \cdot x\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right|} \]

    3. *-commutative 67.5%

       \[\leadsto \left|\color{blue}{x \cdot 2}\right| \cdot \left|\sqrt{\frac{1}{\pi}}\right| \]

    4. rem-exp-log 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right| \]

    5. exp-neg 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\sqrt{\color{blue}{e^{-\log \pi}}}\right| \]

    6. unpow1/2 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}}\right| \]

    7. exp-prod 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}}\right| \]

    8. distribute-lft-neg-out 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{-\log \pi \cdot 0.5}}\right| \]

    9. distribute-rgt-neg-in 67.5%

       \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}}\right| \]

    10. metadata-eval 67.5%

        \[\leadsto \left|x \cdot 2\right| \cdot \left|e^{\log \pi \cdot \color{blue}{-0.5}}\right| \]

    11. exp-to-pow 67.5%

        \[\leadsto \left|x \cdot 2\right| \cdot \left|\color{blue}{{\pi}^{-0.5}}\right| \]

    12. fabs-mul 67.5%

        \[\leadsto \color{blue}{\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|} \]

    13. associate-*r* 67.5%

        \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}\right| \]

    14. rem-square-sqrt 28.4%

        \[\leadsto \left|\color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}}\right| \]

    15. fabs-sqr 28.4%

        \[\leadsto \color{blue}{\sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

    16. rem-square-sqrt 30.1%

        \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

    17. associate-*r* 30.1%

        \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot {\pi}^{-0.5}} \]

13. Simplified 29.9%

    \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}} \]

14. Final simplification 29.9%

    \[\leadsto \frac{x \cdot 2}{\sqrt{\pi}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))